Full-Spectrum Graph Neural Network: Expressive and Scalable

TL;DR

The paper introduces Full-Spectrum GNN, a scalable spectral graph neural network that extends classical models to higher-order signals, improving expressivity especially for heterophilic graphs.

Contribution

It proposes a second-order spectral GNN that lifts signals to node-pair domain and extends spectral filters, with scalable implementations for large graphs.

Findings

FSpecGNN can universally approximate node-pair signals.

It is at most as expressive as Local 2-GNN.

Empirically outperforms on heterophilic benchmarks.

Abstract

It is well established that spectral graph neural networks (GNNs) can universally approximate node signals; however, their expressive power remains bounded by the 1-dimensional Weisfeiler-Lehman test, which is mirrored in their lack of universality for higher-order signals. To go beyond this bound, we propose the Full-Spectrum GNN (FSpecGNN), a second-order generalization of classical spectral GNNs. FSpecGNN advances spectral filtering in two perspectives: (1) it lifts the signal from the node domain to the node-pair domain; and (2) it extends the univariate spectral filter over eigenvalues to a bivariate filter over eigenvalue pairs. We show that classical spectral GNNs arise as a diagonal special case of FSpecGNN, and prove that FSpecGNN can be at most as expressive as Local 2-GNN while universally approximating node-pair signals, the latter being particularly beneficial for heterophilic graph learning. Moreover, FSpecGNN admits scalable implementations that avoid explicit node-pair-level computations; combined with a low-rank approximation that reduces full-spectrum convolution to a combination of polynomial spectral filters, it enables learning on large graphs. Empirically, FSpecGNN validates the predicted expressivity and delivers strong performance on heterophilic benchmarks.